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Equation of a Circle - Worksheet 8 - 10th Grade

Hard

Write and graph equations of circles in standard form (Hard)

23 problems · 20-25 minutes · Skills: circles, coordinate geometry, equations

About This Worksheet

This hard-level worksheet contains 23 problems designed for 10th Grade students practicing equation of a circle. Hard worksheets feature challenging problems including multi-step questions, missing values, and real-world applications to push students further.

The equation of a circle unifies algebra and geometry on the coordinate plane. Knowing how to write, read, and convert circle equations is essential for analytic geometry and appears again in pre-calculus when students study conic sections and in calculus when computing arc lengths.

  • Write the standard form equation of a circle given center and radius
  • Identify the center and radius from a standard-form equation
  • Convert from general form to standard form by completing the square
  • Determine whether a point lies inside, on, or outside a circle using its equation
  • Graph a circle on the coordinate plane given its equation in standard form

How to Solve: Worked Example

Example Problem: Find the centre and radius of x² + y² + 2x - 10y + 25 = 0
Solution Steps:
  1. Group the terms: (x² + 2x) + (y² - 10y) = -25.
  2. Complete the square for x: half of 2 is 1, 1² = 1. For y: half of -10 is -5, (-5)² = 25.
  3. Add both to each side: (x + 1)² + (y - 5)² = -25 + 1 + 25 = 1.
  4. Centre (-1, 5) and radius √1 = 1.
Answer: centre (-1, 5), radius 1
Tip: Whatever you add inside to complete the square must also be added to the right-hand side.
1
Find the centre and radius by completing the square: x² + y² - 2x - 14y - 71 = 0
2
Find the centre and radius by completing the square: x² + y² + 0x + 8y - 9 = 0
3
Find the centre and radius by completing the square: x² + y² + 8x - 14y - 79 = 0
4
Find the centre and radius by completing the square: x² + y² + 12x + 16y + 75 = 0
5
Find the centre and radius by completing the square: x² + y² + 14x - 8y + 40 = 0
6
Find the centre and radius by completing the square: x² + y² - 6x - 4y - 68 = 0
7
Find the centre and radius by completing the square: x² + y² + 0x - 8y + 0 = 0
8
Find the centre and radius by completing the square: x² + y² + 6x + 2y - 111 = 0
9
Find the centre and radius by completing the square: x² + y² - 2x + 14y + 1 = 0
10
Find the centre and radius by completing the square: x² + y² + 10x + 2y + 17 = 0
11
Find the centre and radius by completing the square: x² + y² - 14x - 2y + 41 = 0
12
Find the centre and radius by completing the square: x² + y² - 10x - 4y + 4 = 0
13
Find the centre and radius by completing the square: x² + y² + 14x - 4y - 68 = 0
14
Find the centre and radius by completing the square: x² + y² + 2x - 6y - 111 = 0
15
Find the centre and radius by completing the square: x² + y² - 2x - 4y + 1 = 0
16
Find the centre and radius by completing the square: x² + y² - 6x + 6y + 14 = 0
17
Find the centre and radius by completing the square: x² + y² + 8x + 2y - 8 = 0
18
Find the centre and radius by completing the square: x² + y² - 12x - 2y + 1 = 0
19
Find the centre and radius by completing the square: x² + y² + 0x + 2y - 48 = 0
20
Find the centre and radius by completing the square: x² + y² + 10x - 14y - 26 = 0
21
Find the centre and radius by completing the square: x² + y² - 8x + 6y - 24 = 0
22
Find the centre and radius by completing the square: x² + y² + 14x - 2y + 34 = 0
23
Find the centre and radius by completing the square: x² + y² - 14x - 4y - 91 = 0

Answer Key

1. centre (1, 7), radius 112. centre (0, -4), radius 53. centre (-4, 7), radius 124. centre (-6, -8), radius 55. centre (-7, 4), radius 56. centre (3, 2), radius 97. centre (0, 4), radius 48. centre (-3, -1), radius 119. centre (1, -7), radius 710. centre (-5, -1), radius 311. centre (7, 1), radius 312. centre (5, 2), radius 513. centre (-7, 2), radius 1114. centre (-1, 3), radius 1115. centre (1, 2), radius 216. centre (3, -3), radius 217. centre (-4, -1), radius 518. centre (6, 1), radius 619. centre (0, -1), radius 720. centre (-5, 7), radius 1021. centre (4, -3), radius 722. centre (-7, 1), radius 423. centre (7, 2), radius 12

Frequently Asked Questions

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What skills does this circles worksheet cover?

This worksheet focuses on circles, coordinate geometry, equations. It is designed for 10th Grade students and provides structured practice through varied problem types with a worked example to guide students before they begin.

Can I use this worksheet in my classroom?

Yes. All worksheets on K12Worksheets are free for classroom use. You can print copies for your entire class. With 10 worksheets per subtopic at three difficulty levels, you have plenty of options for differentiated instruction and individualized practice.

How many problems are on this worksheet?

This worksheet contains 23 problems. It typically takes 20-25 minutes to complete. An answer key with all solutions is included so parents and teachers can check work quickly.